THREE CHARACTERIZATIONS OF A SELF-SIMILAR APERIODIC 2-DIMENSIONAL SUBSHIFT

The paper proves both self-similarities and the equality X_φ = Ω_U = X_{P_U,R_U} via explicit desubstitution with marker tiles and Rauzy induction, constructing morphisms α0, α1, α2 and β0, β1, β2 whose compositions equal φ and establishing recognizability; see Theorem 1.1 and its proofs in Sections 4–5, and the concluding summary X_φ = Ω_U = X_{P_U,R_U} (with uniqueness from Exercise 3.12) . By contrast, the candidate solution leaves recognizability for Ω_U incomplete (an unresolved “NE-3/bottom-6” ambiguity) and posits an incorrect commuting relation S∘R_U = R_U∘S, so it does not reach the paper’s main result.